In Shen and Wei (2014) an optimal investment, consumption and life insurance purchase problem
for a wage earner with Brownian information has been investigated. This paper discusses the same
problem but extend their results to a geometric Itô–Lévy jump process. Our modelling framework is
very general as it allows random parameters which are unbounded and involves some jumps. It also
covers parameters which are both Markovian and non-Markovian functionals. Unlike in Shen and Wei
(2014) who considered a diffusion framework, ours solves the problem using a novel approach, which
combines the Hamilton–Jacobi–Bellman (HJB) and a backward stochastic differential equation (BSDE) in
a Lévy market setup. We illustrate our results by two examples.